Chapter Lejeune Dirichlet ( - ) . Types of Relations In this section, we would like to study different types of relations. We know that a relation in a set A is a subset of A × A. Thus, the empty set φ and A × A are two extreme relations.
For illustration, consider a relation R in the set A = { , , , } given by R = {( a , b ): a – b = }. This is the empty set, as no pair ( a , b ) satisfies the condition a – b = . Similarly, R ′ = {( a , b ) : | a – b | ≥ } is the whole set A × A, as all pairs ( a , b ) in A × A satisfy | a – b | ≥ . These two extreme examples lead us to the following definitions.
Definition A relation R in a set A is called empty relation , if no element of A is related to any element of A, i.e., R = φ ⊂ A × A. Definition A relation R in a set A is called universal relation , if each element of A is related to every element of A, i.e., R = A × A. Both the empty relation and the universal relation are some times called trivial relations . Example Let A be the set of all students of a boys school.
Show that the relation R in A given by R = {( a , b ) : a is sister of b } is the empty relation and R ′ = {( a , b ) : the difference between heights of a and b is less than meters} is the universal relation. Solution Since the school is boys school, no student of the school can be sister of any student of the school. Hence, R = φ , showing that R is the empty relation. It is also obvious that the difference between heights of any two students of